What lies beyond the Sedenions

Question

In the construction of types of numbers, we have the following sequence:

$$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$

or:

$$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions} \subset 2^3 \mathrm{-ions} \subset 2^4 \mathrm{-ions} $$

or:

"Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions"

With the following "properties":

• From $\mathbb{R}$ to $\mathbb{C}$ you gain "algebraic-closure"-ness (but you throw away ordering).
• From $\mathbb{C}$ to $\mathbb{H}$ we throw away commutativity.
• From $\mathbb{H}$ to $\mathbb{O}$ we throw away associativity.
• From $\mathbb{O}$ to $\mathbb{S}$ we throw away multiplicative normedness.

The question is, what lies on the right side of $\mathbb{S}$, and what do you lose when you go from $\mathbb{S}$ to one of these objects ?

Answer 1

What you are talking about is precisely the Cayley-Dickson construction.

Remark: I am left wondering what is gained by going past Octonions. The the first 4 are very special as they are the unique 4 normed divison algebras over $\mathbb{R}$. Perhaps someone with more knowledge can point out the possible uses of the Sedenions and their higher counterparts.

Answer 2

You can build tessarines of any dimension of the for $2^n$. They still will be commutative ans associative. This is used in digital signal processing.

Cayley-Dickson's construction, used in quaternions, on the other hand, loses most of its useful properties with the growth of the number of dimensions.